## Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique (2000)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Chen00boundaryknot,

author = {W. Chen},

title = {Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique},

year = {2000}

}

### OpenURL

### Abstract

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity method (DRM), this paper presents an inherently meshless, exponential convergence, integration-free, boundary-only collocation techniques for numerical solution of general partial differential equation systems. The basic ideas behind this methodology are very mathematically simple and generally effective. The RBFs are used in this study to approximate the inhomogeneous terms of system equations in terms of the DRM, while non-singular general solution leads to a boundary-only RBF formulation. The present method is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of non-singular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does no need to introduce the artificial boundary and results in the symmetric system equations under certain conditions. It is also found that the BKM can result in linear analogization formulations of nonlinear partial differential equations with linear boundary conditions if only boundary knots are used. The efficiency and utility of this new technique are validated through a number of typical numerical examples. Some promising developments of the BKM are also discussed.

### Citations

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Citation Context ...e size problem. A number of approaches have been proposed to remedy this problem such as domain decomposition and compactly supported RBFs (CS-RBFs). The latter is very recently developed by Wendland =-=[15]-=-, Wu [16] and Schaback [17]. The CS-RBFs with simple and explicit algebraic form can result in sparse banded matrices and is therefore preferred in many cases. Golberg et al. [2] and Wong et al. [18] ... |

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Citation Context ... approaches have been proposed to remedy this problem such as domain decomposition and compactly supported RBFs (CS-RBFs). The latter is very recently developed by Wendland [15], Wu [16] and Schaback =-=[17]-=-. The CS-RBFs with simple and explicit algebraic form can result in sparse banded matrices and is therefore preferred in many cases. Golberg et al. [2] and Wong et al. [18] respectively applied the CS... |

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Citation Context ... analysis have long been its weakness in handling inhomogeneous terms such as time-dependent and nonlinear problems. The recent introduction of the dual reciprocity BEM (DRBEM) by Nardini and Brebbia =-=[1]-=- eliminates or at least eases to great extent these inefficiencies in the BEM. Notwithstanding, as was pointed out in [2], the method is still more mathematically complex and requires strenuous manual... |

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Citation Context ... by recent works of Chen and Golberg [2-4]. However, their approach [5] to determine analytical approximate solutions is rather complicated and not suitable for general case. By analogy with ideas in =-=[11,12]-=-, we propose a viable alternative scheme to eliminate this problem very easily. In the following, we outline the basic methodology to employ the RBF to approximate particular solution. Let us consider... |

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Citation Context ...d term of Eq. (11) is set up to assure nonsingularity of interpolation matrix, since popular globally support multiquadratics (MQ) and thin plate spline (TPS) are only conditionally positive definite =-=[14]-=-. Note that x k ( ) indicates different independent variables. By forcing Eq. (11) to exactly satisfy Eq. (7) at all nodes together with the d+1 side conditions N+ L ∑ αN+ L+ jxj + αN+ L+ d+ 1 = 0, (1... |

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Citation Context ... [15], Wu [16] and Schaback [17]. The CS-RBFs with simple and explicit algebraic form can result in sparse banded matrices and is therefore preferred in many cases. Golberg et al. [2] and Wong et al. =-=[18]-=- respectively applied the CS-RBFs to the MFS and RBF collocation method successfully. An excellent comprehensive research on this topic can be found in Kansa and Hon [9]. Chen et al [19] recently also... |

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Citation Context ...ucture. 4sThis paper is organized as follows. Section 2 involves the procedure of the DRM and RBF approximation to particular solution. In particular, different from normal approach in Muleskov et al =-=[5]-=-, we use a very simple reverse method to get the analytical approximate particular solution for complicated operators and RBFs. In section 3, we introduce non-singular general solution and derive the ... |

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and M.S.Nerantzaki, The boundary element method for nonlinear problems, Engineering Analysis with Boundary Element
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Citation Context ...l functions of the first kind of the zeroorder. The non-singular general solution of the 3D biharmonic operator is given by ( ) + ( ) * sin λrsinh λ r w = c1 c2 , (61) r r Katsikadelis and Nerantzaki =-=[31]-=- proposed a simple strategy to apply the DRBEM to any partial differential equations which do not include standard Laplace or biharmonic operators. The strategy can be readily extended to the BKM calc... |

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Citation Context ... BEM using time-dependent fundamental solution. The difficulty in applying this scheme may lie in how to satisfy the initial conditions inside domain as in the time-dependent BEM. Recently Zhu et al. =-=[33]-=- used the Laplace transforms to eliminate temporal derivative in the DRBEM solution of time-dependent problems. The accuracy and efficiency of their method are satisfactory. 25sMore recently Golberg a... |

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Citation Context ...r forms are usually rather complicated and are not easy to use. Therefore, we prefer to approximate these inhomogeneous terms by a general numerical approach. Among them, the quasi-Monte Carlo method =-=[13]-=- and RBF in terms of dual reciprocity method [2-5] are two approaches applied successfully. In this paper, we use the latter. The DRM analogizes the particular solution by the use of a series of appro... |

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Boundary element methods using compactly supported radial basis functions
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Citation Context ...d Wong et al. [18] respectively applied the CS-RBFs to the MFS and RBF collocation method successfully. An excellent comprehensive research on this topic can be found in Kansa and Hon [9]. Chen et al =-=[19]-=- recently also used the CS-RBFs to the DRBEM. However, in this study, we will do not use the CS-RBFs to focus on the illustration of the basic idea of the BKM with simpler globally-supported RBFs. Pot... |

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Citation Context ...tem problem with a huge number of nodes. Numerical investigation will be subject of further study along this line. 6. Concluding remarks The present BKM can be regarded one kind of the Trefftz method =-=[36]-=-, where the trial function is required to satisfy governing equation. The BKM distinguishes the other Trefftz techniques in that the RBF is used through non-singular general solution. The BKM also loo... |

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Citation Context ...F to approximate particular solution. Let us consider the differential equation Lux { ( ) }= f( x), x ∈Ω (1) with boundary conditions 5sux ( )= b1 ( x) , x ⊂Γ u , (2) ∂ux ( ) = b ( x) ∂n 2 , x T ⊂Γ , =-=(3)-=- where L is a differential operator, f(x) is a known forcing function. n is the unit outward normal. The above simple forms of boundary conditions are to simplify description. More complicated boundar... |

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(Show Context)
Citation Context ... by recent works of Chen and Golberg [2-4]. However, their approach [5] to determine analytical approximate solutions is rather complicated and not suitable for general case. By analogy with ideas in =-=[11,12]-=-, we propose a viable alternative scheme to eliminate this problem very easily. In the following, we outline the basic methodology to employ the RBF to approximate particular solution. Let us consider... |

2 |
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(Show Context)
Citation Context ...ding u to both sides of Eq. (44), we have 2 2 ∇ u+ u = u− uxx u+ 2+ 2x . (46) Before modeling, we here introduce concept of the Hadmard matrix product to simplify the nonlinear formulation expression =-=[29]-=-. Definition 5.1 Let matrices A=[a ij] and B=[b ij]∈C N×M , the Hadamard product of matrices is defined as A°B= [a ij b ij]∈C N×M . where C N×M denotes the set of N×M real matrices. Following the stan... |

1 |
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(Show Context)
Citation Context ...ranks the best in accuracy [22] and is only the RBF with desirable merit of spectral convergence for general case [4]. However, its accuracy is greatly influenced by the choice of the shape parameter =-=[23,24]-=-. So far the optimal choice of shape parameter is still an open research topic. Despite this problem, the MQ is still most widely used in the RBF solution of various differential systems. For numerica... |

1 |
The multiple reciprocity method of solving transient heat conduction problems
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(Show Context)
Citation Context ...ed to capture the system convection effects accurately in approximation of particular solution, the internal points will not be necessary to improve accuracy. By analogy with multiple reciprocity BEM =-=[30]-=-, it may be feasible to apply the zero-order Bessel function of the first kind repeatedly as the RBF to approximation of particular solution (high-order general solution), which may be superior to the... |